Some aspects of estimation for vector time series models

Abstract

This thesis is primarily concerned with some aspects of estimation for vector autoregressive moving average models which are in their appropriate echelon canonical forms. We restrict attention to the most straightforward part of the modelling procedure, namely, the estimation for fixed values of the Kronecker indices of the structural parameters using ordinary least squares, Gaussian (maximum) likelihood and generalized least squares methods, respectively. Our primary objective here is to give a systematic account of these procedures for handling data and also to provide a thorough exposition of the mathematical details that underlie the techniques. In addition to these abstract mathematical derivations, emphasis will be placed on the practical aspects of the procedures. The discussion of these various issues is organized into six chapters as follows: In Chapter 1 we introduce the class of models and assumptions upon which the results obtained in the thesis are based, and the justification for adopting an echelon structure for such models is also provided. This introductory chapter concludes with a description of the identification procedures for echelon canonical forms. Chapter 2 considers the estimation of the structural parameters using maximum (Gaussian) likelihood procedure and the asymptotic properties of the corresponding estimators are presented. In the evaluation of the parameter estimates, however, explicit expressions are derived for the gradient vector and (approximate) Hessian matrix of the log likelihood function in relatively simple terms. Chapter 3 commences with a procedure for evaluating the least squares estimators. Consistency and asymptotic normality results are established. Chapter 4 assesses the asymptotic relative efficiency of the Gaussian and least squares estim ators via the variance-covariance matrices of the limiting normal distributions obtained in Chapters 2 and 3, respectively. Situations under which substantial loss or gain in efficiency could be expected are discussed and illustrated with some numerical examples. Chapter 5 is devoted to a detailed discussion of the generalized least squares (GLS) procedure for param eter estimation. In particular, the theoretical aspect of the relationship between the GLS and Gaussian estimation methods is investigated and the asymptotic convergence of the GLS estim ator to the Gaussian estim ator is established. Also, an alternative numerical method for implementing the GLS procedure is proposed and some simulation results are presented to illustrate the theory. Finally, in Chapter 6, a method for generating a stable spectral factor from an unstable v x v full rank polynomial operator using closed form algebraic manipulations is proposed. An application of the technique is illustrated and the implementation of the method in the statistical context of system estimation is discussed. Show more